Question

7) Consider the function f(x)=x2e4x.
For this function there are three important intervals: (−∞,A], [A,B], and [B,∞) where A and B are the critical numbers.
Find A
and B
For each of the following intervals, tell whether f(x) is increasing (type in INC) or decreasing (type in DEC).
(−∞,A]:
[A,B]:
[B,∞)

Answer 1

since e^4x is always positive, and

f'(x) = 2x e^4x (2x+1)

f'=0 when x=0 or x = -1/2

f' < 0 where x < -1/2
f' > 0 where -1/2 < x < 0
f' > 0 where x > 0

This is clearly shown at

http://www.wolframalpha.com/input/?i=+x^2+e^%284x%29+for+-1+%3C+x+%3C+0.3